Are there arbitrarily long arithmetic progressions in the sequence of twin primes? II

In an earlier work it was shown that the Elliott-Halberstam conjecture implies the existence of infinitely many gaps of size at most 16 between consecutive primes. In the present work we show that assuming similar conditions not just for the primes but for functions involving both the primes and the Liouville function, we can assure not only the infinitude of twin primes but also the existence of arbitrarily long arithmetic progressions in the sequence of twin primes. An interesting new feature of the work is that the needed admissible distribution level for these functions is just 3/4 in contrast to the Elliott-Halberstam conjecture.     

How small are the increments of partial sums of a discrete random field?

In this paper we establish inferior limit results and path properties for the increments of partial sums of a strictly stationary and linearly positive quadrant dependent (LPQD) or linearly negative quadrant dependent (LNQD) discrete random field with multidimensional indices.     

What is the integral test?

<正>In general,it is not easy to find the exact sum of a series.We could find geometric series with a simple formula for the n-th partial sum S_n.And we are also lucky can find the n-th partial sum S_n of the series ∞∑(n=1)1/(n(n+1)).     

How to prove the integral test?

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A Robin–Dirichlet problem for a nonlinear wave equation with the source term containing a nonlinear integral

In this paper, we consider the Robin–Dirichlet problem for a nonlinear wave equation with the source term containing a nonlinear integral. Using the Faedo–Galerkin method and the linearization method for nonlinear terms, we prove the existence and uniqueness of a weak solution. We also discuss an asymptotic expansion of high order in a small parameter of a weak solution.     

Solution of the operator equationi∈dy/dt=A(t)y on intervals containing turning points

We obtain formal solutions of the equation i∈ dy/dt=A(t)y in the form of complete asymptotic expansions as ∈→0 on intervals containing parabolic or hyperbolic turning points. The highest orders of the power series in ∈ for the formal solutions are studied in detail. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 357–371, March, 2000     

A factorization of an integral 2×2 matrix via a rational method

While a 2×2 integral matrixA with characteristic polynomialx ²-m,m=2,3 (4), and square free, cannot always be expressed as the product of two integral symmetric matrices, it can be expressed as the product of two rational matrices, simultaneously similar to two integral symmetric matrices. This can be achieved in various ways. The paper deals with numerical examples which depend on the 2-part of the ideal class-group attached tox ²-m (see earlier work by the author, especially Acta Arith. 24, 151–156 (1973)).Dedicated to professor E. Hlawka on the occasion of his seventieth birthdayThe term rational is used here also to refer to not algebraic.     

What is the integration by partial fractions?

<正>1.What a rational function?Do you remember what a polynomial function is?If you know a function of the form is a form as following f(x)=anxn+an-ixn-1+...+a1x+a0,where an,an-i,...a1,a0are real numbers and nis nonnegative integer.The domain of a polynomial function is the set of all real numbers.But here are some func-     

A partial improvement of the Ax–Katz theorem

Given a system of polynomial equations over a finite field, estimating the p-divisibility of the number of solutions of the system is a classical problem which has been studied extensively since Chevalley–Warning. The degrees of the polynomials concerned play a crucial role in such estimates. Instead of considering all the variables, we focus on the variables with lower degree and the isolated variables and find a partial improvement of the Ax–Katz theorem. Our result also generalizes, improves and unifies those recently obtained for a single polynomial.     

A new proof of the Askey–Wilson integral via a five-variable Ramanujan’s reciprocity theorem

In this paper, we give an easy and short proof of the well-known Askey–Wilson integral by means of the five-variable Ramanujan’s reciprocity theorem.     

A note on the Moll–Arias de Reyna integral

The Ramanujan Journal - The Moll–Arias de Reyna integral $$\begin{aligned} \int _0^{\infty }\frac{\mathrm{d}x}{(x^2+1)^{3/2}}\frac{1}{\sqrt{\varphi (x)+\sqrt{\varphi (x)}}} \quad \text {...     

A property of the β-Cauchy-type integral with continuous density

  A theorem from the classical complex analysis proved by Davydov in 1949 is extended to the theory of solution of a special case of the Beltrami equation in the z-complex plane (i.e., null solutions of the differential operator ). It is proved that if γ is a rectifiable Jordan closed curve and f is a continuous complex-valued function on γ such that the integral

A new scheme for solving nonlinear Stratonovich Volterra integral equations via Bernoulli’s approximation

In this paper, an efficient method for solving nonlinear Stratonovich Volterra integral equations is proposed. By using Bernoulli polynomials and their stochastic operational matrix of integration, these equations can be reduced to the system of nonlinear algebraic equations with unknown Bernoulli coefficient which can be solved by numerical methods such as Newton’s method. Also, an error analysis is valid under fairly restrictive conditions. Furthermore, in order to show the accuracy and reliability of the proposed method, the new approach is compared with the block pulse functions method by some examples. The obtained results reveal that the proposed method is more accurate and efficient than the block pulse functions method.     

A study on twisted Koecher–Maass series of Siegel cusp forms via an integral kernel

The Ramanujan Journal - The authors wish to draw the attention to a mistake which appears in the proof of Proposition 3 of the above quoted paper [4].     

A note on two Erdősʼs proofs of the infinitude of primes

P. Erdős found numerous theorems, problems, results and conjectures in elementary number theory. Some of them are two Erdősʼs proofs of of the famous Euclidʼs theorem on the infinitude of primes. As noticed below, one of these proofs immediately implies the fact that the number of primes smaller than x is $⩾\log x/(2\log 2)$.     

Predetermined intervals for start times of activities in?the?stochastic project scheduling problem

This paper proposes a new methodology to schedule activities in projects with stochastic activity durations. The main idea is to determine for each activity an interval in which the activity is allowed to start its processing. Deviations from these intervals result in penalty costs. We employ the Cross-Entropy methodology to set the intervals so as to minimize the sum of the expected penalty costs. The paper describes the implementation of the method, compares its results to other heuristic methods and provides some insights towards actual applications.     

A group theory derivation of the relativistic path integral and the “history-string” dynamics

We derive the Feynman path integral for relativistic elementary particles using group theory considerations. We apply an approach in which choosing a symmetry group (or semigroup) allows deriving the kinematics and dynamics of a particle including the state space and the propagator from it. The quantum properties of a particle appear from intertwining two representations of the symmetry (semi)group, one of which describes local properties of the particle and the other describes the particle as a whole. The path-integrallike dynamics appears when the symmetry semigroup has a structure similar to that of the relativistic analogue of the Galilei group (in which the Lorentz-invariant “proper time” plays the role of time) with translations replaced with the semigroup of trajectories (parameterized paths). The classical action in the weight functional of the path integral is defined by this semigroup up to couplings to gauge and/or gravitational fields. The obtained formalism is suitable for describing not only pointlike particles but also nonlocal objects of the “history-string” type, which, as previously shown, allow explaining quark confinement.